100 A SHORT HISTORY OF SCIENCE 



(2) That the convex surface of a segment of a sphere is equal 

 to the area of a circle whose radius is equal to the straight line 

 from the vertex of the segment to any point in the perimeter of 

 its base. 



(3) That the cylinder having a great circle of the sphere for its 

 base and the diameter of the sphere for its altitude exceeds the 

 sphere by one-half, both in volume and in surface. It was the 

 figure for this last proposition which was at his wish carved upon 

 his tombstone. 



In attempting to solve the problem of passing a plane 

 through a sphere so that the segments thus formed shall have 

 either their surfaces or their volumes in an assigned ratio, he is 

 led to a cubic equation ; he appears to have given both a solution 

 and a criterion for the existence of a positive root, but the work 

 is lost. 



In his Conoids and Spheroids he deals with the bodies formed 

 by the revolution of the ellipse, parabola, and hyperbola, by means 

 of plane cross-sections, ascertains the volume of these solids by 

 comparing the portion between two neighboring planes with an 

 inscribed and a circumscribed cylinder, much in the modern 

 manner. 



It is not possible to find in all geometry more difficult and more 

 intricate questions or more simple and lucid explanations (than those 

 given by Archimedes). Some ascribe this to his natural genius; 

 while others think that incredible effort and toil produced these, to 

 all appearance, easy and unlabored results. No amount of inves- 

 tigation of yours would succeed in attaining the proof, and yet, once 

 seen, you immediately believe you would have discovered it ; by so 

 smooth and so rapid a path he leads you to the conclusion required. 



- Plutarch. 



In other branches of mathematical science than geometry the 

 work of Archimedes was relatively even more important. 



The so-called Cattle Problem, for example, is a notable per- 

 formance in the algebra of linear equations. 



" The sun had a herd of bulls and cows, all of which were either 



